Continuous-time quantum walk - Revision history

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2014-12-13T13:19:31Z

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	~~[[Image:Desargues theorem alt~~.~~svg\|thumb\|right\|300px\|Two perspective triangles, with their perspective axis~~ and ~~center]]~~		Prοper diet is the primary factor in good condition. This may keep the mind and body ѕensation fantastic. You need to take in properly to feel happy. The following аre ways to affect yоur particular diet program.<br><br>ϒou need to combine 600 mg to 900 milligrams of garlic into wҺat yοu eat. Garlic cloves is really a organic presсription Videoѕ de vigrx plus [[http://tecate.redbeardtwo.com/vigrx-plus-reviews-fda-simple-techniques-for-a-nutritious-diet-plan/ Tecate.redbeardtwo.com]] antibiotic, and will fight a ѵariety of ailments for example heart disease and cancеrs. Also, it is an all-natural antі--yеast ɑnd anti--microbe whiϲh can help your bodily orgɑns. You ought to take in cloves and garlic clߋves each day.<br><br>It's essential to pаy ɑttention to meals labeling. 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	~~Two figures~~ in a [[~~plane (geometry)\|plane]] are '''perspective from~~ a ~~[[point (geometry)\|point]]''' ''O'' if the lines joining corresponding points of the figures all meet at ''O''~~. ~~[[Duality (projective geometry)\|Dually~~]], ~~the figures are said to be '''perspective from~~ a ~~line''' if the points~~ of ~~intersection of corresponding lines all lie on one line~~. ~~The proper setting for this concept~~ is ~~in [[projective geometry]] where there will be no special cases due to parallel lines since~~ all ~~lines meet~~. ~~Although stated here for figures~~ in ~~a plane, the concept is easily extended to higher dimensions~~.

	~~==Terminology==~~
	~~The line which goes through the points where the figure~~'s ~~corresponding sides intersect is known~~ as ~~the '''axis of perspectivity''', '''perspective axis''', '''homology axis'''~~, or ~~archaically, '''perspectrix'''~~. ~~The figures~~ are ~~said to be perspective from this axis~~. ~~The point at which~~ the ~~lines joining the corresponding vertices of the perspective figures intersect~~ is ~~called the '''center of perspectivity'''~~, '~~''perspective center''', '''homology center''', '''pole''', or archaically '''perspector'''. The figures are said to be perspective from this center~~.<~~ref~~>~~{{harvnb\|Young\|1930\|loc=p. 28}}~~<~~/ref~~>

	~~==Perspectivity==~~
	~~If each of~~ the ~~perspective figures consists of all the points on~~ a ~~line (a [[projective range\|range]]) then transformation of~~ the ~~points of one range~~ to the ~~other is called~~ a ~~''central perspectivity''~~. ~~A dual transformation~~, ~~taking all~~ the ~~lines through a point (a [[pencil (mathematics)\|pencil]])~~ to ~~another pencil by means of an axis of perspectivity is called an ''axial perspectivity''~~.<~~ref~~>~~{{harvnb\|Young\|1930\|loc=p. 29}}~~<~~/ref~~>

	~~==Triangles==~~
	~~An important special case occurs when the figures are [[triangle]]s~~. ~~Two triangles that are perspective from~~ a ~~point are called a ''central couple'' and two triangles~~ that are ~~perspective from a line are called an ''axial couple''~~.~~<ref>{{harvnb\|Dembowski\|1968\|loc=p~~. ~~26}}<~~/~~ref>~~

	==~~Notation~~==
	~~[[Karl von Staudt~~]~~] introduced the notation <math>ABC \doublebarwedge abc</math>~~ to ~~indicate that triangles ABC and abc are perspective~~.~~<ref>~~[[H. S. ~~M. Coxeter]] (1942) ''Non~~-~~Euclidean Geometry'', [[University~~ of ~~Toronto Press]~~], ~~reissued 1998 by [[Mathematical Association of America]]~~, ~~ISBN 0~~-~~88385-522-4 . 21,2~~.<~~/ref~~>

	~~==Related theorems and configurations==~~
	~~[[Desargues' theorem]] states that~~, ~~a central couple of triangles is axial~~. ~~The converse statement, an axial couple of triangles is central, is equivalent (either~~ can ~~be used~~ to ~~prove~~ the ~~other)~~. ~~Desargues' theorem~~ can ~~be proved~~ in ~~the [[real projective plane]~~], and ~~with suitable modifications for special cases~~, in ~~the [[Euclidean plane]]~~. ~~[[Projective plane]]s in which this result~~ can be ~~proved~~ are ~~called ''Desarguesian planes~~''.

	~~There~~ are ~~ten points associated with these two kinds of perspective: six~~ on ~~the two triangles, three on the axis~~ of ~~perspectivity~~, ~~and one at the center of perspectivity~~. ~~[[Projective duality\|Dually]]~~, ~~there~~ are ~~also ten lines associated with two perspective triangles: three sides of the triangles~~, ~~three lines through the center of perspectivity~~, and the ~~axis of perspectivity~~. ~~These ten points~~ and ~~ten lines form an instance of~~ the ~~[[Desargues configuration]]~~.

	~~[[File:Pappus hexagon.svg\|thumb\|240px\|Two triply-perspective triangles ''BbY'' and ''CcX'']]~~
	~~If two triangles are a central couple in~~ at ~~least two different ways (with two different associations of corresponding vertices~~, ~~and two different centers of perspectivity) then they are perspective~~ in ~~three ways~~. ~~This is one of~~ the ~~equivalent forms of [[Pappus's hexagon theorem\|Pappus's (hexagon) theorem]]~~.<~~ref~~>~~{{harvnb\|Coxeter\|1969\|loc=p. 233 exercise 2}}~~ <~~/ref~~> ~~When this happens,~~ the ~~nine associated points (six triangle vertices and three centers) and nine associated lines (three through each perspective center) form an instance of~~ the ~~[[Pappus configuration]]~~.

	~~The [[Reye configuration]] is formed by four quadruply-perspective tetrahedra in an analogous way~~ to the ~~Pappus configuration~~.

	~~== Notes ==~~
	~~{{reflist}}~~

	~~==References==~~
	* {{Citation \| last1=Coxeter \| first1=Harold Scott MacDonald \| author1-link=Harold Scott MacDonald Coxeter \| title=Introduction to ~~Geometry \| publisher=[[John Wiley & Sons]] \| location=New York \| edition=2nd \| isbn=978-0-471-50458-0 \| mr=123930 \| year=1969}}~~
	* {{Citation \| last1=Dembowski \| first1=Peter \| title=Finite geometries \| publisher=[[Springer-Verlag]] \| location=Berlin, ~~New York \| series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]]~~, ~~Band 44 \| mr=0233275 \| year=1968 \| isbn=3-540-61786-8}}~~
	* {{citation\|first=John Wesley\|last=Young\|title=Projective Geometry\|year=1930\|publisher=Mathematical Association of ~~America\|series=The Carus Mathematical Monographs (#4)}}~~

	~~[[Category:Triangles]]~~
	~~[[Category:Projective geometry]]~~

169.229.140.28: /* External links */

2014-01-31T21:54:07Z

External links

← Older revision		Revision as of 23:54, 31 January 2014
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	~~If you~~ are ~~looking for~~ a ~~specific plugin, then you can just search for~~ the ~~name~~ of the ~~plugin~~. ~~Good luck on continue learning how~~ to ~~make~~ a ~~wordpress website. This CMS has great flexibility to adapt various extensions and add-ons. Out~~ of ~~the various designs~~ of ~~photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it~~. The ~~top 4 reasons~~ to ~~use Business Word - Press Themes~~ for a ~~business website are:~~. ~~<br><br>~~		[[Image:Desargues theorem alt.svg\|thumb\|right\|300px\|Two perspective triangles, with their perspective axis and center]]
			Two figures in a [[plane (geometry)\|plane]] are '''perspective from a [[point (geometry)\|point]]''' ''O'' if the lines joining corresponding points of the figures all meet at ''O''. [[Duality (projective geometry)\|Dually]], the figures are said to be '''perspective from a line''' if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in [[projective geometry]] where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions.

	~~As you know today Word - Press has turn out~~ to be ~~a tremendously popular open source publishing and blogging display place~~. ~~You may either choose to link only to~~ the ~~top-level category pages~~ or ~~the ones that contain information about your products and services~~. ~~We also help~~ to ~~integrate various plug-ins to expand~~ the ~~functionalities~~ of the ~~web application~~. ~~Furthermore~~, ~~with~~ the ~~launch~~ of ~~Windows 7 Phone~~ is the ~~smart phone market nascent App~~. ~~Once you~~'~~ve installed the program you can quickly begin by adding content~~ and ~~editing it with features such as bullet pointing, text alignment and effects without having to do all~~ the ~~coding yourself.~~ <br><br>~~The entrepreneurs can easily captivate their readers by using these versatile themes~~. ~~After sending these details, your Word - Press blog will be setup within a few days~~. ~~This platform can be customizedaccording to the requirements of the business~~. ~~If you have any type~~ of ~~questions pertaining to where and the best ways to use~~ [~~http://test~~.~~mike5810~~.~~com~~/~~blogs/post/12413 wordpress dropbox backup~~], ~~you could contact us at~~ the ~~web page~~. ~~Newer programs allow website owners~~ and ~~internet marketers to automatically and dynamically change words~~ in ~~their content to match~~ the ~~keywords entered by their web visitors~~ in ~~their search queries~~'~~a feat that they cannot easily achieve~~ with ~~older software. If you've hosted your Word - Press website~~ on ~~a shared hosting server then it'll be easier for you to confirm~~ the ~~restricted access to your site files. <br><br>Word - Press installation is very easy~~ and ~~hassle free. But the Joomla was created as~~ the ~~CMS over years~~ of ~~hard work~~. ~~Next you'll go by way~~ of ~~to your simple Word - Press site. A whole lot worse~~, ~~your site will likely be useless as well as your merchandise won't sell if no one has~~ the ~~endurance to wait for~~ the ~~web pages to load~~. ~~The popularity~~ of ~~Word - Press has increased~~ the ~~demand for Word~~ - ~~Press themes~~ and ~~these themes sells like hot cake on~~ the ~~internet~~. <br><br>~~Website security has become a major concern among individuals all over~~ the ~~world~~. ~~Being a Plugin Developer, it~~ is ~~important for you~~ to ~~know that development of Word~~ - ~~Press driven website should be done only when you enable debugging. However~~, ~~you must also manually approve or reject comments so that your website does not promote parasitic behavior. Page speed is an important factor in ranking~~, ~~especially with Google. Get started today so that people searching for your type~~ of ~~business will be directed to you.~~		==Terminology==
			The line which goes through the points where the figure's corresponding sides intersect is known as the '''axis of perspectivity''', '''perspective axis''', '''homology axis''', or archaically, '''perspectrix'''. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the '''center of perspectivity''', '''perspective center''', '''homology center''', '''pole''', or archaically '''perspector'''. The figures are said to be perspective from this center.<ref>{{harvnb\|Young\|1930\|loc=p. 28}}</ref>

			==Perspectivity==
			If each of the perspective figures consists of all the points on a line (a [[projective range\|range]]) then transformation of the points of one range to the other is called a ''central perspectivity''. A dual transformation, taking all the lines through a point (a [[pencil (mathematics)\|pencil]]) to another pencil by means of an axis of perspectivity is called an ''axial perspectivity''.<ref>{{harvnb\|Young\|1930\|loc=p. 29}}</ref>

			==Triangles==
			An important special case occurs when the figures are [[triangle]]s. Two triangles that are perspective from a point are called a ''central couple'' and two triangles that are perspective from a line are called an ''axial couple''.<ref>{{harvnb\|Dembowski\|1968\|loc=p. 26}}</ref>

			==Notation==
			[[Karl von Staudt]] introduced the notation <math>ABC \doublebarwedge abc</math> to indicate that triangles ABC and abc are perspective.<ref>[[H. S. M. Coxeter]] (1942) ''Non-Euclidean Geometry'', [[University of Toronto Press]], reissued 1998 by [[Mathematical Association of America]], ISBN 0-88385-522-4 . 21,2.</ref>

			==Related theorems and configurations==
			[[Desargues' theorem]] states that, a central couple of triangles is axial. The converse statement, an axial couple of triangles is central, is equivalent (either can be used to prove the other). Desargues' theorem can be proved in the [[real projective plane]], and with suitable modifications for special cases, in the [[Euclidean plane]]. [[Projective plane]]s in which this result can be proved are called ''Desarguesian planes''.

			There are ten points associated with these two kinds of perspective: six on the two triangles, three on the axis of perspectivity, and one at the center of perspectivity. [[Projective duality\|Dually]], there are also ten lines associated with two perspective triangles: three sides of the triangles, three lines through the center of perspectivity, and the axis of perspectivity. These ten points and ten lines form an instance of the [[Desargues configuration]].

			[[File:Pappus hexagon.svg\|thumb\|240px\|Two triply-perspective triangles ''BbY'' and ''CcX'']]
			If two triangles are a central couple in at least two different ways (with two different associations of corresponding vertices, and two different centers of perspectivity) then they are perspective in three ways. This is one of the equivalent forms of [[Pappus's hexagon theorem\|Pappus's (hexagon) theorem]].<ref>{{harvnb\|Coxeter\|1969\|loc=p. 233 exercise 2}} </ref> When this happens, the nine associated points (six triangle vertices and three centers) and nine associated lines (three through each perspective center) form an instance of the [[Pappus configuration]].

			The [[Reye configuration]] is formed by four quadruply-perspective tetrahedra in an analogous way to the Pappus configuration.

			== Notes ==
			{{reflist}}

			==References==
			* {{Citation \| last1=Coxeter \| first1=Harold Scott MacDonald \| author1-link=Harold Scott MacDonald Coxeter \| title=Introduction to Geometry \| publisher=[[John Wiley & Sons]] \| location=New York \| edition=2nd \| isbn=978-0-471-50458-0 \| mr=123930 \| year=1969}}
			* {{Citation \| last1=Dembowski \| first1=Peter \| title=Finite geometries \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 \| mr=0233275 \| year=1968 \| isbn=3-540-61786-8}}
			* {{citation\|first=John Wesley\|last=Young\|title=Projective Geometry\|year=1930\|publisher=Mathematical Association of America\|series=The Carus Mathematical Monographs (#4)}}

			[[Category:Triangles]]
			[[Category:Projective geometry]]

en>Jacksonwalters: /* External links */ Added link to a CTQW on Wolfram Demonstrations.

2012-08-15T20:05:05Z

External links: Added link to a CTQW on Wolfram Demonstrations.

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